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An advection-diffusion-reaction size-structured fish population dynamics model combined with a statistical
parameter estimation procedure: Application to the Indian Ocean skipjack tuna fishery
Blaise Faugeras, Olivier Maury
To cite this version:
Blaise Faugeras, Olivier Maury. An advection-diffusion-reaction size-structured fish population dy-
namics model combined with a statistical parameter estimation procedure: Application to the Indian
Ocean skipjack tuna fishery. Mathematical Biosciences and Engineering, AIMS Press, 2005, 2 (4),
pp.719-741. �hal-00496619�
Volume2, Number4, October2005 pp.1–23
AN ADVECTION-DIFFUSION-REACTION SIZE-STRUCTURED FISH POPULATION DYNAMICS MODEL COMBINED WITH A
STATISTICAL PARAMETER ESTIMATION PROCEDURE:
APPLICATION TO THE INDIAN OCEAN SKIPJACK TUNA FISHERY
Blaise Faugeras
CNRS I3S,
Les Algorithmes, 2000 route des lucioles, BP 121, 06903, Sophia Antipolis Cedex, France
Olivier Maury
Institut de Recherche pour le D´eveloppement,
Centre de Recherche Halieutique, avenue Jean Monnet, BP 171, 34200 S`ete, France
(Communicated by ??)
Abstract. We develop an advection-diffusion size-structured fish population dynamics model and apply it to simulate the skipjack tuna population in the Indian Ocean. The model is fully spatialized, and movements are parame- terized with oceanographical and biological data; thus it naturally reacts to environment changes. We first formulate an initial-boundary value problem and prove existence of a unique positive solution. We then discuss the numer- ical scheme chosen for the integration of the simulation model. In a second step we address the parameter estimation problem for such a model. With the help of automatic differentiation, we derive the adjoint code which is used to compute the exact gradient of a Bayesian cost function measuring the dis- tance between the outputs of the model and catch and length frequency data.
A sensitivity analysis shows that not all parameters can be estimated from the data. Finally twin experiments in which pertubated parameters are recovered from simulated data are successfully conducted.
1. Introduction. Fish population dynamics models together with parameter es- timation techniques are essential to provide assessment of the fish abundance and fishery exploitation level. Their use forms the basis of scientific advice for fisheries managements. This is particularly true for tuna fisheries, which are among the most valuable in the world and subject to increasing fishing pressure and to the effects of climate changes.
Discrete age-structured models with crude representations of space are most of the time used for fisheries stock assessments [1, 2]. The classical data used in fishery science to calibrate models are fishing effort, catch and length frequency data.
Length frequency data are not straightforward to use. Fish of the same age can exhibit very different sizes depending of their history [3, 4]. Therefore, to compare
2000Mathematics Subject Classification. 92D25, 92D40, 86A05, 35K15, 35K20, 35K57, 65M06, 86A22, 65K10, 93B30.
Key words and phrases. Population dynamics model, size structure, well-posed initial- boundary value problem, statistical parameter estimation, tuna fisheries, stock-assessment.
1
the outputs of age-structured models with length frequency data, a Gaussian size distribution is generally added to each age class. However, because of non-uniform mortality over sizes, bias on growth and mortality estimates may result from this procedure [5].
Another point concerning tuna fisheries is that they are highly heterogeneous in space and time. This has a significant effect on their functioning. Important mi- grations of fish occur at various scales, so that fish movements have to be explicitly represented using spatialized models [6].
These are some of the main problems of current stock assessment models. It is necessary to carry on the modelling effort by proposing and testing more complex models dealing more accurately with size distributions and spatial heterogeneity.
This paper follows this direction and its purpose is twofold.
First we describe in section 2 a model of population dynamics in which both size and space are taken as structure variables to account for growth, movements of fish, environmental variability and variable distribution of fishing effort. The model con- sists of an advection-diffusion-reaction equation. Spatial advection-diffusion models have a long history in ecology [7, 8, 9], but their use in fishery science has grown recently, particularly for tuna population modeling purposes [10, 11, 6]. To model tropical tuna population in the Indian Ocean realistically, our model needs to re- flect the heterogeneous distribution and movements of the population linked to the environment and fishing effort heterogeneity. Thus in the model fish movements depend on oceanographical and biological data through a habitat suitability index.
Recruitment, that is to say the input of young fishes in the model, is modeled as a source term involving a nonlocal nonlinearity. The structure of the model enables a direct and simultaneous comparison with the two main types of data available for tuna fisheries: catches and size frequencies.
We assess the mathematical well-posedness of the model in section 3. We for- mulate an initial-boundary value problem, introduce a variational formulation and show existence of a unique weak solution. As often with nonlinear problems the proof uses a fixed-point argument. We also show the positivity of the solution.
Our second goal is to develop and test a data assimilation procedure to estimate the parameters of the model in a realistic skipjack tuna fishing simulation. Indeed one of the main objectives of tuna population modeling is to provide robust evalu- ations of stocks which are hardly possible nowadays for tuna fisheries in the Indian Ocean because of the lack of robust estimations of many biological parameters, such as natural and fishing mortality rates or recruitment parameters. Section 4 deals with the numerical implementation of the simulation model. Then in Section 5 we describe the data assimilation method developed for parameter estimation as well as the Bayesian likelihood approach used to formulate a cost function measuring the distance between the outputs of the model and the data. The paper ends with some numerical experiments conducted in section 6 to validate the algorithm in the case of the Indian Ocean skipjack fishery.
2. The model. The dynamics of the population of fish is described through a density function p(x, y, s, t), where position (x, y)∈Ω the bounded domain repre- senting the ocean, size or length s∈(S0, S1) and timet ∈(0, T). The number of fish of size between s1ands2 at timet and position (x, y) is given by the integral
Z s2
s1
p(x, y, s, t)ds.
The population density follows an advection-diffusion process in space. Let D(x, y, s, t) =diag(Du(x, y, s, t), Dv(x, y, s, t))
be the space diffusion matrix and
V(x, y, s, t) = (u(x, y, s, t), v(x, y, s, t))T
be the velocity field. The population density also follows an advection-diffusion process in the size variable (see section 2.1 for more details). Let d(s) denote the dispersion coefficient in size and γ(s) be the growth rate. Finally let m(s) and F(x, y, s, t) denote the natural and fishing mortality rates, and R(x, y, s, t, p) the recruitment source term (see sections 2.4 and 2.3). The density functionpfollows the balance law,
∂tp = div(D∇p)−div(V p) +∂s(d∂sp)−∂s(γp)
−(m+F)p+R(p), in Ω×(S0, S1)×(0, T),
(1) where∇ and div are the usual differential operators on Ω.
This equation has to be completed with initial conditions
p(x, y, s,0) =p0(x, y, s), ∀(x, y, s)∈Ω×(S0, S1) (2) and boundary conditions
∂sp(x, y, S0, t) =∂sp(x, y, S1, t) = 0, ∀(x, y, t)∈Ω×(0, T), (3) and
∇p(x, y, s, t)·n(x, y) = 0, in∂Ω, ∀(s, t)∈(S0, S1)×(0, T), (4) where n(x, y) is the unit normal vector pointing outside Ω. Homogeneous Neu- mann boundary conditions ats=S0 and s=S1 express the fact that the size of individuals can not reach values lower thanS0 or larger thanS1.
The parameterizations of the processes involved in the time evolution of the population are described in detail in the following subsections.
2.1. Movements: Advection-diffusion in space. Diffusion and velocity in space have a physical and a biological component. The biological components depend on a habitat suitability index function,hsi(x, y, t), and its first space derivatives. The index hsi depends on temperature, T(x, y, t), and forage, F ood(x, y, t) which are input data for the model. The biotic affinities for these environmental factors are defined as
fT(x, y, t) = 1/(1 +exp(−αT(T(x, y, t)−T0))), (5) and
fF ood(x, y, t) =F ood(x, y, t)/(KF ood+F ood(x, y, t)). (6) All parameters are given in Table 1, and Fig. 2 shows plots offT andfF ood. In its most general form the index is defined as
hsi(x, y, t) = (fT(x, y, t))pT(fF ood(x, y, t))pF ood. (7) The velocity field is computed as
V(x, y, s, t) =Vphy(x, y, t) +Vhsi(x, y, s, t), (8)
Table 1. Habitat suitability index parameters name value unit
αT 0.35 (degree C)−1
T0 20 degree C
KF ood 1000 J
pT 1
pF ood 1
where Vphy = (uphy, vphy)T represents the physical velocity (computed by a hy- drodynamical model) andVhsi= (uhsi, vhsi)T represents biological velocity defined by
uhsi(x, y, s, t) =uhsi0(1−hsi(x, y, t))( ∂xhsi(x, y, t)
khsi+|∂xhsi(x, y, t)|)( s S1
), vhsi(x, y, s, t) =uhsi0(1−hsi(x, y, t))( ∂yhsi(x, y, t)
khsi+|∂yhsi(x, y, t)|)( s S1
).
(9) Vhsi is proportional to length (large fish can swim faster than small ones) and to (1−hsi)∇hsi(the model transports the population towards the most suitable places for fish living according to the temporal habitat index evolution). The diffusion coefficients are defined as follows:
Du(x, y, s, t) = Dmin+ (Dmax−Dmin)
×(1−hsi(x, y, t))
×(1− |∂xhsi(x, y, t)|
khsi+|∂xhsi(x, y, t)|)( s S1
)2, Dv(x, y, s, t) = Dmin+ (Dmax−Dmin)
×(1−hsi(x, y, t))
×(1− |∂yhsi(x, y, t)|
khsi+|∂yhsi(x, y, t)|)(s S1
)2.
(10)
The interpretation of such a parameterization is similar to the one given for Vhsi
and all parameters are given in Table 2.
Table 2. Movements parameters
name value unit
Dmin 104 m2.s−1 Dmax 105 m2.s−1
uhsi0 10 m.s−1
khsi 2.5×10−7 m−1
2.2. Growth and dispersion in size. As time goes on and fish grow older, their size or length increases with a growth rate γ(s) (see Eq. (11) and Table 3). A diffusion term in the size variable with a dispersion rated(s) (see Eq. (12) and Table 3) is included to account for individuals having the same age but different sizes.
Indeed, in a fish population individuals of the same age can often differ markedly in size [4]. This variability in growth can result from many mechanisms, including genetic or behavorial traits that confer different performances to individuals, and factors such as environmental heterogeneity and variability [3]. In fishery science,
this variability is usually taken into account in age-structured models using a length- at-age relation perturbed by a Gaussian noise (see, for example, [12]). The model discussed here is size-structured and uses a diffusion term in the size variable with dispersion rate d(s) to account for individuals having the same age but different sizes [13, 5]. The advection-diffusion term in size can be seen as the limit of a random walk model in which each individual grows with an average velocity but has at each time step a small binomial probability to grow faster or slower than this average (see [8] for more details). We consider that fish growth follows a Von Bertalanfy curve:
γ(s) =γ1−γ2( s−S0
S1−S0
), (11)
d(s) =d1+d2γ(s) γ1
. (12)
Table 3. Parameters for growth and dispersion in size
name value unit
γ1 3.858×10−9 m.s−1 γ2 3.858×10−9 m.s−1 d1 3.215×10−12 m2.s−1 d2 3.215×10−12 m2.s−1
2.3. Recruitment. Recruitment is computed as a function of the stock spawning biomass,
B(x, y, t, p) = Z S1
smat
f r(s)w(s)p(x, y, s, t)ds, (13) wheresmatis the minimum size at maturity, the fecundity rate f r(s) is given by
f r(s) = bf
(1 +exp(−af(s−smat)), (14) and the weightw(s) of a fish of sizesby
w(s) =awsbw. (15)
We use a Beverton and Holt [3] stock-recruitment relation and obtain, R(x, y, s, t, p) = l1(S0,sr)(s) b0B(x, y, t, p)
kB+B(x, y, t, p), (16) where l1(S0,sr) is the usual characteristics function andsr is the maximum size of recruitment.
2.4. Natural and fishing mortality. The mortality rate is split into size-dependent natural mortality m(s) [14] and a fishing mortality rate F(x, y, s, t). The fishing mortality rate is defined as the sum of the Nf fishing mortality induced by each fleet,
F(x, y, s, t) =
Nf
X
f=1
Ff(x, y, s, t). (17)
The mortality rate induced by each fleet is described by the following equation:
Ff(x, y, s, t) =qf(s)Ef(x, y, t), (18)
Table 4. Recruitment parameters
name value unit
bf 0.5
af 1 m−1
smat 0.5 m
aw 4.82×10−6 kg.m−bw bw 3.36
b0 0.35×10−9 m−1.s−1 kB 0.5×10−4 kg
sr 0.5 m
where qf(s) is the size-dependent catchability coefficient for fleet f (that is the probability for a fish of sizesto be caught by a unit of fishing effort of fleetf), and Ef(x, y, t) is the observed fishing effort.
3. Mathematical well-posedness. In this section we prove existence and unique- ness of a positive weak solution to the model.
3.1. Functional spaces. Let us introduce some functional spaces. The study is conducted on the open set Q = Ω×(S0, S1). Let T < ∞ be a fixed time.
The population density, p, is considered as an element of the functional space H = L2(Q), whose Hilbert space machinery is convenient to use. H is equipped with the scalar product
(p, q)H = Z
Ω
Z S1
S0
pqdsdxdy,
and we denote by||.||H the induced norm. We also consider the separable Hilbert space defined byH1=H1(Q) and equipped with the scalar product
(p, q)H1 = Z
Ω
Z S1
S0
(pq+∇p.∇q+∂sp∂sq)dsdxdy.
We denote by||.||H1 the induced norm onH1.
We will also have to consider the Banch spaceL∞=L∞(Q ×(0, T)) equipped with the norm
||p||∞=inf{M,|p(x, y, s, t)| ≤M a.e. in Q ×(0, T)}.
L2(0, T, H) is the space of functions L2 in time with values in H, equipped with the norm,
||p||L2(0,T,H)= [ Z T
0
||p(t)||2Hdt]1/2,
andL∞(0, T, H) is the space of functionsL∞ in time with values in H, equipped with the norm,
||p||L∞(0,T,H)=inf{M,||p(t)||H≤M a.e in(0, T)}.
SimilarlyC([0, T], H) is the space of continuous functions on [0, T] with values in H. Further,C([0, T], H),L∞(0, T, H) andL2(0, T, H) are Banach spaces.
ClassicallyH′denotes the dual ofHand (H1)′the dual ofH1. WhenHis identified with its dual, we have the scheme
H1⊂H =H′⊂(H1)′,
where each space is dense in the following and the imbeddings are continuous.
Let us denote byW(H1) the Hilbert space W(H1) ={p∈L2(0, T, H1),dp
dt ∈L2(0, T,(H1)′)}.
Lemma 3.1. Every p∈W(H1)is a.e equal to a continuous function from[0, T]to H. Moreover we have the following continuous imbedding,
W(H1)⊂C([0, T], H).
Proof. See, for example, Dautray and Lions [15]
3.2. Assumptions on the data and preliminary transformation of the sys- tem. The mortality rates are assumed to satisfy
• m(s), F(x, y, s, t)≥0 a.e inQ ×(0, T),m, F ∈L∞.
If we assume that the input temperature and forage fields,T(x, y, t) andF ood(x, y, t), are positive and regular enough, it appears clearly from section 2 that
• Du(x, y, s, t), Dv(x, y, s, t)≥Dmin>0, a.e in Q ×(0, T),Du, Dv∈L∞;
• u(x, y, s, t), v(x, y, s, t) are differentiable with respect toxandy, respectively andu, v, ∂xu, ∂yv∈L∞.
It is also clear from section 2 that
• d(s)≥d1>0, a.e in (S0, S1),d∈L∞;
• γ(s) is differentiable with respect tos, and γ, ∂sγ∈L∞;
• f r(s), w(s)≥0 a.e in (S0, S1),f r, w∈L∞.
We also assume that the initial distributionp0(x, y, s) satisfies
• p0(x, y, s)≥0 a.e in Q,p0∈H.
To prove our existence-uniqueness result, it is convenient to perform a change of unknown function: p satisfies (1)-(4) if and only if ˆp=e−λtpis a solution to the same equations where−(m+F)pis replaced with −(m+F+λ)pin Eq. (1) and the recruitmentR(x, y, s, t, p) is replaced by
R(x, y, s, t,ˆ p) = lˆ 1[S0,sr](s) b0e−λtB(x, y, t,ˆ p)ˆ
kBe−λt+ ˆB(x, y, t,p)ˆ , (19) B(x, y, t,ˆ p) =ˆ
Z S1
smat
f r(s)w(s)ˆp(x, y, s, t)ds. (20) In the remaining part of the mathematical analysis, this change of unknown is implicitly done and we omit the ˆp notation. The constant λ will be fixed to a convenient value below. Moreover, the possible nullification of the termkBe−λt+ B(x, y, t,ˆ p) invites us to defineˆ
R(x, y, s, t, p) = l1[S0,sr](s) b0e−λtB(x, y, t, p)
kBe−λt+|B(x, y, t, p)|, (21) This formulation is being used in the following. We will show that if the initial dis- tribution,p0is nonnegative thenp≥0 a.e. inQ ×(0, T); thus the two formulations are equivalent.
3.3. Variational formulation.
3.3.1. The bilinear form a(t, p, q). Formally multiplying Eq. (1) by a function q and integrating by parts onQleads to the definition of the following bilinear form.
Forp, q∈H1 let us define a(t, p, q) =
Z
Q
D∇p∇qdxdyds+ Z
Q
V.∇pqdxdyds +
Z
Q
d(∂sp)(∂sq)dxdyds+ Z
Q
γ(∂sp)qdxdyds +
Z
Q
(m+F+λ+ div(V) +∂sγ)pqdxdyds.
(22)
Lemma 3.2. For a.e. t ∈ (0, T), a(t, p, q) is continuous on H1 ×H1, and for λ large enough, a(t, p, q) is coercive on H1. There exist two constants C1 > 0 andC2>0, depending on ||D||∞,||d||∞,||u||∞,||v||∞,||γ||∞,||∂xu||∞,||∂yv||∞,
||∂sγ||∞,||F||∞,||m||∞,Dmin,d1 andλ, such that
|a(t, p, q)| ≤C1||p||H1||q||H1, ∀p, q∈H1, (23) a(t, p, p)≥C2||p||2H1, ∀p∈H1. (24) Proof. The proof is classical and we omit it.
3.3.2. The nonlinear operatorR. In this section we show that the recruitment term R(x, y, s, t, , p) (cf. Eqs. (13)-(16)) allows us to define a Lipschitz continuous oper- atorRonL2(0, T, H).
Lemma 3.3. Let
Λ = b0(S1−S0)||f r||∞||w||∞
kB
; then the application
p(x, y, s, t)7→R(x, y, s, t, p)
defines a bounded nonlinear operator, R, Lipschitz continuous fromL2(0, T, H)to L2(0, T, H)with Lipschitz constant Λ.
Proof. Let us first notice that the application (t, B) 7→ h(t, B) = b0e−λtB kBe−λt+|B|
from [0, T]×RtoRsatisfies
|h(t, B)| ≤ b0
kB
|B|. (25)
Furthermoreh(t, B) is Lipschitz continuous inB uniformly int∈[0, T],
|h(t, B1)−h(t, B2)| ≤ b0
kB
|B1−B2|, ∀B1, B2∈R, ∀t∈[0, T]. (26) Since B(x, y, t, p)2 = (
Z S1
smat
f r(s)w(s)p(x, y, s, t)ds)2, we obtain using Cauchy- Schwarz
B(x, y, t, p)2≤(S1−S0)||f r||2∞||w||2∞||p(x, y, ., t)||2L2(S0,S1). (27) Hence from (25) and (27) we deduce that∀t∈[0, T],
||Rp(t)||2H = Z
Ω
Z S1
S0
[ l1[S0,sr](s)h(t, B(x, y, t, p))]2dxdyds,
≤Λ2||p(t)||2H,
(28)
and thatRis well-posed onL2(0, T, H).
In the same way, if top1(resp. p2) we associateB1(resp. B2), we deduce from (26) that∀t∈[0, T],
||Rp1(t)− Rp2(t)||2H = Z
Ω
Z S1
S0
[ l1[S0,sr](s)(h(t, B1(x, y, t, p1))
−h(t, B2(x, y, t, p2)))]2dxdyds,
≤Λ2||p1(t)−p2(t)||2H,
(29)
and thusRis Lipschitz continuous onL2(0, T, H).
3.3.3. Weak solutions. We can now give the definition of a weak solution to system (1)-(4).
Definition 3.1. We say thatp∈W(H1), is a weak solution of system (1)-(4) if
∀q∈H1, (dp
dt, q)H+a(t, p, q) = (Rp, q)H, (30) in theD′(]0, T[) sens,
andp(0) =p0.
Then we can prove the following result.
Theorem 3.1. System (1)-(4) admits a unique non-negative weak solution.
Proof.
Existence and uniqueness. The proof consists mainly in defining a nonlinear operator Θ by freezing the nonlinear term Rp and applying Banach fixed-point theorem to Θ. The fixed point is the desired solution.
Step 1. Let ˆp be fixed inW(H1) and in Eq. (30) let us replace (Rp, q)H by (Rˆp, q)H. The problem becomes linear inpand admits a unique solution (e.g, [15]).
This solution defines an operator Θ onW(H1), Θˆp=p.
Step 2. Let us show that for T sufficiently small Θ satisfies the following properties:
1. Θ leaves invariant the ball,
Br={p∈W(H1), ||p||L∞(0,T,H)≤r, r≥ ||p0||H
p(1−(Λ2T /2C2))} that is, ΘBr⊂Br.
Takingq=pas test function in (30), integrating on [0, t], using the coercive- ness ofaand Cauchy-Schwarz inequality, we obtain
Z t 0
1 2
d
dt||p(σ)||2H+C2||p(σ)||2H1dσ≤ Z t
0
||Rˆp(σ)||H||p(σ)||dσ.
For allα >0, Young inequality leads to
||p(t)||2H+ 2C2
Z t 0
||p(σ)||2H1dσ≤ Z t
0
1
α||Rˆp(σ)||2Hdσ+ Z t
0
α||p(σ)||2dσ+||p0||2H, and choosingα= 2C2 gives
||p(t)||2H≤ 1 2C2
Z t 0
||Rˆp(σ)||2Hdσ+||p0||2H.
Then using Eq. (28) we obtain
||p(t)||2L∞(0,T,H)≤Λ2T 2C2
||p||ˆ 2L∞(0,T,H)+||p0||2H. If||ˆp||L∞(0,T,H)≤rthen||p||L∞(0,T,H)≤rfor Λ2C2T
2r2+||p0||2H ≤r2that is to sayr2(1−Λ2C2T
2)≥ ||p0||2H. This implies Λ2C2T
2 <1 which is valid for smallT, andr≥ ||p0||H
p(1−(Λ2T /2C2)).
2. Θ is a strict contraction onBr, there exists 0< k <1 such that∀p1, p2∈Br,
||Θp1−Θp2||L∞(0,T,H)≤k||p1−p2||L∞(0,T,H).
Letp1= Θˆp1andp2= Θˆp2. Substracting the two associated Eq. (30), taking p1−p2as test function and again using the coerciveness ofa, Cauchy-Schwarz and Young inequality leads to
d
dt||p1(t)−p2(t)||2H ≤ 1 2C2
||Rˆp1(t)− Rˆp2(t)||2H. Sincep1(0) =p2(0) =p0, we deduce
||p1(t)−p2(t)||2H ≤ 1 2C2
Z t 0
||Rˆp1(σ)− Rˆp2(σ)||2Hdσ, and
||p1−p2||2L∞(0,T,H)≤ Λ2T 2C2
||ˆp1−pˆ2||2L∞(0,T,H). Then for Λ2C2T2 <1, Θ is a strict contraction.
Step 3. For T small enough, by Banach-fixed point theorem Θ admits a unique fixed point which is the desired solution on (0, T). SinceT does not depend onp0, the same procedure can be applied on (T,2T), ... until a solution is found on the desired time interval.
Positivity. Let p1 ≥ 0 be given in W(H1), and let us define the sequence (pn)n≥1 by Θpn=pn+1. Let us prove thatp2is non-negative:
Takingp−2 =max(0,−p2) as test function in (30) leads to (d
dtp2, p−2)H+a(t, p2, p−2) = (Rp1, p−2)H, and therefore to
1 2
d
dt||p−2||2H≤ 1 2
d
dt||p−2||2H+a(t, p−2, p−2) =−(Rp1, p−2)H, Sincep1≥0 then Rp1≥0 and−(Rp1, p−2)H ≤0. It results that
d
dt||p−2||2H≤0, that is,
||p−2(t)||2H ≤ ||p−2(0)||2H=||p0−||2H = 0,
andp2≥0. An induction then shows thatpn≥0, ∀n≥1, and since the sequence converges to the solutionp, this latter is non-negative.
4. Numerical treatment of the model. In the approximation procedure of the model a centered finite difference discretization is used. Equation (1) is solved on a grid with a spatial resolution of 2 degrees, i.e. ∆y = 120 nautical miles in the latitudinal direction and ∆x = ∆ycos(θ) in the longitudinal direction (θ is the latitude angle), a discrete length step, ∆s of 4 cm, and a discrete time step
∆t of one day is used. The discretization points are denoted by (xi, yj, sl, tn) with i ∈ [1 : I], j ∈ [1 : J] (assuming here for simplicity that the domain Ω is rectangular), l ∈ [1 : L] and n ∈ [1 : N]. In what follows, pni,j,l denotes the numerical approximation ofp(xi, yj, sl, tn).
Several difficulties arise in the computation of the solution to Eq. (1). First the numerical scheme has to be very stable because of possible strong variations in space and time of the advection and diffusion coefficients, u, v, Du, Dv. Moreover the numerical solution of Eq. (1) is to be used in a numerical function minimiza- tion procedure to obtain estimates of model parameters. Therefore the solution algorithm must be fast because the model and its adjoint may have to be solved hundreds of time. Moreover, the function minimization algorithm may test param- eter values that do not necessarily guarantee numerical stability.
The selected scheme combines a splitting method [16, 17] and the use of the MUSCL scheme for advection terms (monotonic upstream centered scheme for con- servation laws [18]). At each time step, given an approximationpn ofp(x, y, s, tn), the computation of pn+1 from pn is achieved through four steps. The advection- diffusion equation in thexvariable is integrated first, on [tn, tn+1]:
∂tp(x, y, s, t) =∂x(Du(x, y, s, t)∂xp)−∂x(u(x, y, s, t)p),
p(x, y, s, tn) =pn. (31)
It results in a first approximationpn+1,1. Then the advection-diffusion equation in they variable is integrated on [tn, tn+1] starting from pn+1,1:
∂tp(x, y, s, t) =∂y(Dv(x, y, s, t)∂yp)−∂y(v(x, y, s, t)p),
p(x, y, s, tn) =pn+1,1. (32)
It results in a second approximationpn+1,2. Then the advection-diffusion term in thesvariable is integrated on [tn, tn+1] starting frompn+1,2:
∂tp(x, y, s, t) =∂s(d(s)∂sp)−∂s(γ(s)p),
p(x, y, s, tn) =pn+1,2. (33)
It results in a third approximationpn+1,3. Finally mortality and recruitment are integrated on [tn, tn+1] starting frompn+1,3:
∂tp(x, y, s, t) =−(m(s) +F(x, y, s, t))p+R(x, y, s, t, p),
p(x, y, s, tn) =pn+1,3. (34)
It results in the final valuepn+1.
In each of the first three steps, diffusion is treated implicitly in time, and the MUSCL scheme is used for the advection term. For example, the discretization used to solve Eq. (31) can be written as follows:
−2∆x∆t2(Du;i−1,j,ln+1 +Du;i,j,ln+1 )pn+1i−1,j,l
+(1 +2∆x∆t2(Dn+1u;i−1,j,l+ 2Dn+1u;i,j,l+Du;i+1,j,ln+1 )pn+1i,j,l
−2∆x∆t2(Du;i,j,ln+1 +Du;i+1,j,ln+1 )pn+1i+1,j,l
=∆x∆t(fi+1/2,j,ln −fi−1/2,j,ln ),
(35)
where fi+1/2,j,ln =
uni+1/2,j,l(pni,j,l+ 1/2∆ni(1−uni+1/2,j,l∆x∆t)), if uni+1/2,j,l≥0, uni+1/2,j,l(pni+1,j,l−1/2∆ni+1(1 +uni+1/2,j,l∆x∆t)), otherwise.
(36) In this last equation, if (pni−1,j,l≤pni,j,l≤pni+1,j,l)
or if (pni+1,j,l≤pni,j,l≤pni−1,j,l) then
∆min= min(|p
n
i+1,j,l−pni−1,j,l|
2 ,2|pni+1,j,l−pni,j,l|,2|pni,j,l−pni−1,j,l|),
∆ni =sign(pni+1,j,l−pni−1,j,l)∆min, (37) and otherwise,
∆ni = 0. (38)
5. Parameter estimation. In this section we describe the data assimilation al- gorithm developed to estimate the parameters of the model.
5.1. The outputs of the model corresponding to the data. Total catches in weight as well as length frequencies of the catches are computed and compared to observations to estimate the parameters of the model. In each cell (i, j) of the grid, where during monthm(30 days), the fishing effort is nonzero, catches of fleetf are computed as follows:
Ci,j,m,f =
L
X
l=1
30m
X
n=30(m−1)+1
qf,lEf,i,j,ln pni,j,lwl∆s∆x∆y∆t, (39) and length frequencies as
Qi,j,l,m,f=
30m
X
n=30(m−1)+1
qf,lEf,i,j,ln pni,j,l∆s∆x∆y∆t
L
X
l=1
30m
X
n=30(m−1)+1
qf,lEnf,i,j,lpni,j,l∆s∆x∆y∆t
. (40)
5.2. The cost function. The parameters of the model are denoted in what follows by K ∈ RNp where Np is the number of parameters. K is being estimated in a Bayesian context by computing the mode of the posterior density function of the parameters knowing the data. We use the maximum of posterior distribution method [19], which involves minimizing the sum of the negative log-likelihood of the data plus the log of prior density functions.
We assume that the observation errors for catch data follow a log-normal distri- bution. Therefore the contribution of total catches to the negative log-likelihood
is
JC(K) = 1 2σ2C
X
i,j
X
m
X
f
(log(Ci,j,m,f)−log(Ci,j,m,fobs ))2. (41) The observation errors for length frequency data are assumed to be normal and the contribution of frequency data to the negative log-likelihood reads
JQ(K) = 1 2σ2Q
X
i,j
X
l
X
m
X
f
(Qi,j,l,m,f−Qobsi,j,l,m,f)2. (42) The negative log of prior density functions for the parameters is
JP(K) =X
n
1
2σn2(Kn−Kn0)2, (43) where Kn0 are the reference a priori parameters given in Tables 1-4. The cost function to be minimized is the sum of those three terms:
J(K) =JC(K) +JQ(K) +JP(K). (44) The parameters have different units and orders of magnitude. To avoid any nu- merical difficulties that might arise from this during the minimization, we adimen- sionalize the parameter vector K, dividing each parameter Ki by its first guess a priori value Ki0. Let D = diag(Ki0) then the adimensionalized control vector is k=D−1K. Such an adimensionalization procedure can be regarded as a precon- ditioning for the minimization. The final cost function is
j(k) =j(D−1K) =J(K), (45)
and the a priori reference adimentionalized parameter vector isk0= 1.
5.3. Optimization: Computing the gradient with the adjoint model. To minimize the cost functionj, we used the quasi-Newton algorithm implemented in then1qn3 Fortran subroutine of Gilbert and Lemar´echal [20]. The computation of the gradient of j with respect to control variables is required at each step of the minimization. This gradient results in one integration of the adjoint model. The adjoint code was obtained using the automatic differentiation program Odyss´ee [21, 22], which is an efficient tool for deriving adjoint codes since it enables the automatic production of adjoint instructions. However, codes produced by auto- matic differentiation do not usually use computer memory in a very efficient way.
Saving the direct model trajectory is the major problem. A differentiation pro- gram has to follow systematic methods to provide the evaluation trajectory. Thus Odyss´ee systematically uses a local calculation and storage technique for the trajec- tory. Automatically differentiating a 3D model and using the adjoint code directly seems impossible for the moment. Thus the code generated by Odyss´ee had to be improved manually. A Taylor test was then conducted to compare the exact deriva- tives computed by the adjoint code to a finite difference approximation. Generally speaking, one aims at verifying that
r(ǫ) = j(k+ǫδk)−j(k) ǫ(∇j(k), δk) −→
ǫ→00 (46)
for any direction of perturbationδk. We present in Table 5 the result of such a test.
Asǫ becomes smaller, one observes that the ratior(ǫ) first tends linearly towards 1 up to ǫ = 10−6, which is the optimal value for a finite difference computation.
Afterwards, the substraction of close floating-point numbers leads to a large cancel- lation error, which dominates the truncation error coming from the computation of
Table 5. Result of a Taylor test
ǫ r(ǫ)
10−1 1.010756398 10−2 1.001203422 10−3 1.000115118 10−4 1.000015861 10−5 1.000001463 10−6 1.000000518 10−7 0.999993143 10−8 0.999985731 10−9 0.999847627 10−10 0.997712868
the gradient by the finite difference method. A Taylor test with such a numerical behavior of the ratio r(ǫ) is said to be correct. It verifies that the adjoint code provides an exact computation of the gradient.
6. Numerical results.
6.1. The simulation set-up. The standard run consists in a one-year simulation for the Indian Ocean. The spatial numerical grid used is shown on Fig. 1. Sizes of simulated skipjack tunas range fromS0= 0.4 mtoS1= 1.2m.
Figure 1. The 46×32 numerical grid used to integrate the population dynamics model on the Indian Ocean (earth in black, ocean in white).
Inititial conditions are chosen to be homogeneous over the space grid with a size distribution
p0(x, y, s) = 0.1e−0.5s. (47)
This distribution assumes that the population is dominated by small organisms.
Using these initial conditions a spin-up run of 6 years is conducted in order to reach an experimental and numerical fixed-point where mortality processes balance
the recruitment process and the total biomass slowly varies around a mean value during the year. The final distribution at the end of the spin up period provides the initial distribution for the standard run.
Since skipjack tunas inhabit the surface layer of the ocean, the model is forced with monthly velocity of oceanic surface currents, sea surface temperature and for- age fields (Fig. 2). Velocity and temperature fields are outputs of the ocean general circulation model OPA,1whereas forage fields are outputs of a size structured model representing the energy flow in marine ecosystems from zooplankton to organisms of the size of tuna forage [23].
0 0,05 0,1 0,15 0,2 0,25
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 temperature (degree celcius)
data frequency
0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
response function value
0 0,05 0,1 0,15 0,2 0,25 0,3
0 10000 20000 30000 40000 50000 60000 70000 80000 90000 10000 11000 12000 13000 14000
forage
data frequency
0 0,2 0,4 0,6 0,8 1
response function value
Figure 2. Response functions, fT and fF included in the hsi formulation corresponding to temperature (left) and forage (right).
Frequency distributions over the whole grid and the whole year of temperature and forage values are also plotted.
Figure 3 shows a contour plot of the index hsi for the month of march. The strong south-north gradient of the index in the lower half of the map is typical of the area. Low temperatures are not suitable for tuna, which stay in the upper half of the map as shown on Fig. 4.
To test the possibility of estimating some parameters of the model from standard fishing data, we conduct in the following section numerical experiments with a syn- thetic data set computed by one simulation of the model. All parameters are set to their reference a priori values. Moreover for the sake of simplicity the two mortal- ity parametersmandqare assumed to be size-independent; that is constants with valuesm= 4.2438 10−8s−1[6] andq= 6.43 10−8s−1. Only one fleet is considered (Nf = 1), and the fishing effort is assumed to be constant and homogeneous during the year on an area roughly corresponding to the real fishing areas of the purse seine fleet in the Indian Ocean (see Fig. 5). With this configuration, a data set is computed following Eqs. (39) and (40).
6.2. Sensitivity analysis. We conduct a sensitivity analysis in order to identify the most important input parameters whose changes impact the most the outputs of the model (catches and length frequencies).
1http://www.lodyc.jussieu.fr/opa/
0.4 0.65 0.6
0.75 0.7 0.8 0.85 0.9 0.9
0.2
Figure 3. Contour plot of the indexhsifor the month of March.
1e−0141e−012 1e−010
1e−008 1e−006 1e−005 0.0005 0.001
0.003 0.002
0.0005
0.003 0.001 0.0002
2e−005
0.0005
0.0001
Figure 4. Contour plot of the population density function summed over size classes at the end of the month of March.
As a measure of the outputs of the model we consider the quantities hC(k) =X
i,j
X
m
(Ci,j,m)2 (48)
and
hQ(k) =X
i,j
X
l
X
m
(Qi,j,l,m)2. (49)
Then the vector of relative sensitivities of these quantities to variations of the input parameters computed at the pointk are
sC(k) = ∇hC(k)
hC(k) (50)
and
sQ(k) = ∇hQ(k)
hQ(k) . (51)
Figure 5. Distribution of the fishing efforts used in the simula- tions. F = 0 in the gray area andF = 1 in the black area.
Each of these relative sensitivity vector requires the computation of a gradient which is easily obtained with one integration of the adjoint model.
Table 6 shows the relative sensitivity vectors,sC(k0) andsQ(k0), computed with the initial a priori parameter vectork0. Globally the different relative sensitivities are quite low, indicating that it may be difficult to estimate correctly all the param- eters of the model using the two types of fishery data which are generally available.
In the remaining part of this paper we will not try to estimate parameters which have the lowest sensitivities (10−3, 10−2). These parameters are
• Dmin,Dmax related to diffusion in space,
• d1,d2,γ1 andγ2 related to growth,
• af andsmatrelated to recruitment.
Interestingly the relative sensitivities sQ corresponding to variations in bf, kB
andaw(see Table 6) are exactly equal up to a sign. This comes from the formulation of recruitment in the model, which from Eqs. (13)-(16) can be rewritten with obvious notations as
b0B
kB+B = b0bfawBˆ
kB+bfawBˆ = b0Bˆ kB
bfaw
+ ˆB
. (52)
An inconsistency in the formulation of the inverse parameter estimation problem appears clearly. The 3 parameters,bf,kB andaw can not be determined indepen- dently, since for example an increase in kB can also be interpreted as a decrease in bf or in aw. For this reason in our identification experiments we keep bf and aw fixed to their reference values and only try to estimatekB. Moreover since the length/weight parameters aw and bw are well known we also do not select bw for the parameter estimation formulation.
Although the 2 mortality parametersm and q do not correspond to very high sensitivities, we will try to estimate them since they really are badly known.
Finally the chosen formulation includes 11 parameters to be estimated:
• movements parameters : khsi, pT, pF ood, αT, T0, KF ood, uhsi0
• recruitment parameters : b0, kB
• mortality parameters : m, q
Table 6. Relative sensitivities of catch and length frequency data param. sC(k0) sQ(k0)
Dmin 1.39 10−2 −8.21 10−3 − Dmax 7.81 10−2 −1.12 10−2 − khsi 1.36 10−1 −3.59 10−2 + pT −1.54 10−1 4.75 10−2 + pF ood −1.06 10−1 1.50 10−2 + αT −1.00 3.12 10−1 + T0 2.58 10−1 −8.06 10−2 + KF ood −1.01 10−1 1.43 10−2 + uhsi0 −1.68 10−1 4.08 10−2 + d1 5.53 10−3 −3.83 10−2 − d2 4.78 10−3 −3.47 10−2 − γ1 3.87 10−2 −3.51 10−1 − γ2 −4.93 10−3 3.36 10−2 − b0 1.93 10−1 3.02 10−1 + bf −1.55 10−1 −2.49 10−1 − kB 1.55 10−1 2.49 10−1 + aw 3.19 10−1 2.49 10−1 −
bw 4.16 3.24 −
af −1.22 10−3 −2.07 10−3 − smat −3.90 10−2 −6.27 10−2 − m −9.11 10−2 3.42 10−2 + q 7.47 10−2 2.49 10−2 +
Note: In the last column, a + or−indicates whether or not the corresponding parameter is estimated.
6.3. Identification experiments. An essential validation step to perform before assimilation of real observed data is to conduct twin experiments. Synthetic data are produced by the model using the first guess parameter vectork0. To fully test the possibility of recovering the selected parameters from the synthetic data, no penalty term is added and the cost function reduces to
j(k) =jC(k) +jQ(k). (53)
The assumed variances are as follows: σC = 0.1 and σQ = 0.01. This provides a good balance between the two terms of j. In the experiments conducted, the convergence criterion is ||∇j(k)||
||∇j(k0)|| ≤ǫ, whereǫis a small value fixed to 10−5. 6.3.1. Experiment 1. A first numerical experiment was conducted to assess the capacity of the parameter estimation algorithm to distinguish between low (or high) recruitment and high (or low) mortality rates on the one hand and natural and fishing mortality rates on the other hand. Therefore in this optimization only the four parametersb0,kB,m, andqcan vary, the others being fixed to their reference a priori value used to simulate the data. Different first guesses for the parameter vector were obtained by perturbing these four parameters within reasonable range (up to 50% of their reference value). All the corresponding optimizations converged
to the minimum of the cost function. The results of such an experiment are shown in Figs. 6, 7, and 8. The convergence criterion is satisfied after 20 iterations. The cost function value decreased from 2.9 105to 5.1 10−7indicating that it has reached its global minimum and all 4 parameters have been recovered.
0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40
0 5 10 15 20
iteration number
parameter value
Figure 6. Convergence of the 4 selected parameters towards their reference value (k0= 1) during the optimization experiment 1.
1.00E-07 1.00E-05 1.00E-03 1.00E-01 1.00E+01 1.00E+03 1.00E+05
0 5 10 15 20
iteration number
cost function
Figure 7. Evolution of the cost function j(k) during the opti- mization experiment 1.
6.3.2. Experiment 2. A second numerical experiment was conducted to assess the capacity of the parameter estimation algorithm to recover all the 11 parameters at the same time. Therefore in this second optimization experiment all of 11 param- eters can vary. Different first guesses for the parameter vector were obtained by perturbing these parameters within reasonable range (up to 20% of their reference
1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06
0 5 10 15 20
iteration number
gradient
Figure 8. Evolution of the gradient ||∇j(k)|| during the opti- mization experiment 1.
0.90 0.95 1.00 1.05 1.10 1.15 1.20
0 10 20 30 40 50 60
iteration number
parameter value
Figure 9. Convergence of the 11 selected parameters towards their reference value (k0= 1) during the optimization experiment 2.
value). All the corresponding optimizations converged to the minimum of the cost function. The results of such an experiment are shown in Figs. 9, 10, and 11.
The convergence criterion is satisfied after 59 iterations. The cost function value decreased from 6.4 106to 2.0 10−1, indicating that it has reached its global minimum and all 11 parameters have been recovered.
1.00E-01 1.00E+00 1.00E+01 1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07
0 10 20 30 40 50 60
iteration number
cost function
Figure 10. Evolution of the cost functionj(k) during the opti- mization experiment 2.
1.00E+03 1.00E+04 1.00E+05 1.00E+06 1.00E+07 1.00E+08 1.00E+09
0 10 20 30 40 50 60
iteration number
gradient
Figure 11. Evolution of the gradient||∇j(k)|| during the opti- mization experiment 2.
